Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
2
votes
1 answer

What is the degree of the map $f_k: S^1 \mapsto S^1: (\cos(2\pi t), \sin(2\pi t))\mapsto (\cos(2k\pi t), \sin(2k\pi t))$

Let $S_1$ be the unit sphere in $\mathbb{R}^2$ and define the following map: $$f_k: S^1 \mapsto S^1: (\cos(2\pi t), \sin(2\pi t))\mapsto (\cos(2k\pi t), \sin(2k\pi t)).$$ We are asked to calculate the degree of this map. De degree of $f$ is defined…
emiel647
  • 81
2
votes
1 answer

$f_*: \pi_1(\mathbb{C} \backslash \{-1, 0, 1\}; 2) \rightarrow \pi_1(\mathbb{C} \backslash \{0, 1\}; 4)$ is injective

Let $f: \mathbb{C}\backslash \{-1, 0, 1\} \rightarrow \mathbb{C} \backslash \{0, 1\}$ be defined by $f(z) = z^2$. Show that the homomorphism $f_*: \pi_1(\mathbb{C} \backslash \{-1, 0, 1\}; 2) \rightarrow \pi_1(\mathbb{C} \backslash \{0, 1\}; 4)$ is…
Math_Day
  • 1,227
2
votes
1 answer

Fundamental group of the quotient space of the disk obtained by identifying points on the boundary that are 120 degree aparts

Let $X$ be the quotient space of the disk, $\{(x,y)\in \mathbb R^{2} \ | \ x^{2}+y^{2}\leq 1 \}$, obtained by identifying points on the boundary that are $120$ degrees apart. How can we find the fundamental group of $X$ ?
hebele
  • 31
2
votes
0 answers

Homology group of complement of embedded curve in a sphere.

I know the answer for the question if the embedding in the question is from $I=[0,1]$ into $\mathbb{S}^2$ or from $\mathbb{S}^1$ into the sphere. It is direct from the Proposition 2B.1 from Chapter 2B of Hatcher Algebraic Topology. However, I want…
Jose Pérez Cano
  • 137
2
votes
0 answers

Topological space whose fundamental group is the circular group

I have the following problem: Find a space $X$ path-connected, whose universal covering is contractible and is such that $\pi_{1}(X)=\mathbb{S}^{1}$, where $\mathbb{S}^{1}\subseteq\mathbb{C}$ and it is considered a group under multiplication. A…
Fernando Mauricio Rivera Vega
  • 445
2
votes
1 answer

Counterexample to naive excision theorem

The excision theorem states that under suitable conditions, given subspaces $Z\subset A\subset X$, then we have isomorophisms of homology groups $H_n(X-Z,A-Z)\rightarrow H_n(X,A)$ for all $n$. Intuitively, it seems to me that since relative chain…
Vasting
  • 2,055
2
votes
1 answer

Hatcher Question Long Exact Homology

There is a sentence in Hatcher that I am really struggling to understand. We have the following theorem If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact…
Vasting
  • 2,055
2
votes
1 answer

Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.

Let $D$ denote the closed unit disc in the plane with boundary the unit circle $S^1$. Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.
kiwi
  • 249
2
votes
0 answers

finding the generator for the local homology $H^n(X,X-\{x\})$

I was reading Hatcher's algebraic topology book,I found when computing something,it's convenient to give the generator explictly,hence I was tring to figure out what is the generator for the local homology $H^n(X,X-\{x\})$ for a manifold $X$ with…
yi li
  • 4,786
2
votes
0 answers

Homology groups of $\mathbb{R}^3$ relative to a disjoint union of two copies of $S^1$

Problem #5: Let $C_1$, $C_2$ be two copies of $S^1$ disjointly embedded in $\mathbb{R}^3$. Compute $H_i(\mathbb{R}^3,C_1\cup C_2)$ for all $i\in\mathbb{N}$. If I am understanding this correctly, this problem seems really easy. However, I'm not…
user82668
  • 43
2
votes
0 answers

Proof of that SO(3) is not simply connected.

I want to prove that $\pi_1(SO(3))\cong \mathbb{Z}/2\mathbb{Z}$. I have already proved that there exists a surjection $\mathbb{Z}/2\mathbb{Z}\rightarrow \pi_1(SO(3))$.So I want to show that $\pi_1(SO(3))\neq 0$, but I still haven't come up with how…
user53216
2
votes
0 answers

Doubt in exercise 2.2.9 of Hatcher's Algebraic Topology

The exercise is the following : Compute the Homology of the quotient space of $S^1 \times S^1$ obtained by identifying points in the circle $S^1\times \{x_0\}$ that differ by $2\pi/m$ rotation and identifying points in the circle $\{x_0\}\times…
Someone
  • 4,737
2
votes
1 answer

What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus?

What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus? I would greatly appreciate any help that you could give me.
picklekong
  • 103
2
votes
1 answer

what is the meaning of $q$ times a generator of $\pi_1(S^1)?$

I have some confusion about the statement in Allen hatcher Book Page No:$33$ Theorem $1.10:$ For every contnious map $f:S^2 \to \mathbb{R}^2$ there exist a pair of antipodal points $x$ and $-x $ in $S^2$ with $f(x)=f(-x)$ In the theorem of the…
jasmine
  • 14,457
2
votes
2 answers

What is meaning of $\pi_0(X,A, x_0)$

I'm studying Algebraic Topology, and I'm thinking about the meaning of $\pi_0(X, A, x_0)$. In Hatcher's exercise, $\pi_0(X, A, x_0):= \pi_0(X, x_0) / \pi_0 (A, x_0)$. But, I don't know how to quotient these set. $\pi_0 (A, x_0)$ is path component of…
probafds123
  • 402
1 2 3
…
99 100